Exercise 28, § 3

Let G be a topological group and let X be the Banach space of bounded continuous mappings of G into $\mathbf{C}$ endowed with the norm

For each $f \in X$ and each $s \in G$, let $U_s f$ denote the function $t \to f(s^{-1}t)$, which belongs to X, so that the $U_s$, as s runs through G, form a group of isometries $G'$ of X. An element $f \in X$ is said to be an almost periodic function (on the left) on G if f is an almost periodic element of X with respect to the group $G'$. For this to be so it is necessary and sufficient that, for each $\varepsilon > 0$, there should exist a finite number of elements $s_i \in G$ with the property that, for each $s \in G$, there is at least one index i such that $|f(s^{-1}t) - f(s_i^{-1}t)| \leq \varepsilon$ for all $t \in G$.

a) Suppose that f is almost periodic. Let Y be the closure in X of the orbit of f under $G'$, and let $\Gamma$ be the closure in $C_u(Y; Y)$ of the set of restrictions $V_s$ of the $U_s$ to Y ($s \in G$). For each $\sigma \in \Gamma$, let $f_\sigma \in Y$ be the image of f under $\sigma$. For each $s \in G$, we have

Deduce that there is a continuous function $\bar{f}$ on $\Gamma$ such that $f(s) = \bar{f}(V_s)$ [use Exercise 27 c)].

b) Show by using a) that a function $f \in X$ is almost periodic on G if and only if it is of the form $g \circ \varphi$, where $\varphi$ is the canonical mapping of G into the compact group $G^c$ associated with G (Set Theory, Chapter IV, § 3, no. 3, Example 8) and g is a continuous mapping of $G^c$ into $\mathbf{C}$.

c) Take G to be the additive group $\mathbf{R}$. Show that if f is almost periodic on $\mathbf{R}$, then for each $\varepsilon > 0$ there exists a real number T > 0 such that every interval of $\mathbf{R}$ of length T contains a number s such that

(s is called an “ε-period” of f ). [Use b) and Exercise 2 of Chapter V, § 1.]